Optical vortices evolving from helicoidal integer and fractional phase steps

The evolution of a wave starting at z = 0a s exp (iαφ) (0 φ 0as trength n optical vortex, whose neighbourhood is described in detail. Far from the axis, the wave is the sum of expi (αφ + kz) and a diffracted wave from r = 0. The paraxial wave and the wave far from the vortex are incorporated into a uniform approximation that describes the wave with high accuracy, even well into the evanescent zone. For fractional α, n o fractional-strength vortices can propagate; instead, the interferenc eb etween an additional diffracted wave, from the phase step discontinuity, with expi (αφ + kz) and the wave scattered from r = 0, generates a pattern of strength-1 vortex lines, whose total (signed) strength Sα is the nearest integer to α. F or small|α − n|, t heselines are close t ot hez axis. As α passes n +1 /2, Sα jumps by unity, so a vortex is born. The mechanism involves an infinite chain of alternating-strength vortices close to the positive x axis for α = n +1 /2, which annihilate in pairs differently when α> n +1 / 2a nd when α< n +1 /2. There is a partial analogy between α and the quantum flux in the Aharonov–Bohm effect.